On function spaces on symmetric spaces

TL;DR

This paper investigates the behavior of smooth vectors in L^p spaces on semisimple symmetric spaces, showing they tend to zero at infinity, which advances understanding of harmonic analysis on these spaces.

Contribution

It establishes that smooth vectors for the regular representation on L^p spaces on symmetric spaces vanish at infinity, a new result in harmonic analysis.

Findings

Smooth vectors in L^p(Y) vanish at infinity.

Advances understanding of harmonic analysis on symmetric spaces.

Provides new insights into the structure of function spaces on G/H.

Abstract

Let Y=G/H be a semisimple symmetric space. It is shown that the smooth vectors for the regular representation of G on L^p(Y) vanish at infinity.